Abstract
For integers \(2 \le t \le k\), we consider a collection of k set families \({\mathcal {A}}_j: 1 \le j \le k\) where \({\mathcal {A}}_j = \{ A_{j,i} \subseteq [n] : 1 \le i \le m \}\) and \(|A_{1, i_1} \cap \cdots \cap A_{k,i_k}|\) is even if and only if at least t of the \(i_j\) are distinct. In this paper, we prove that \(m =O(n^{ 1/ \lfloor k/2 \rfloor })\) when \(t=k\) and \(m=O(n^{1/(t-1)})\) when \(2t-2 \le k\) and prove that both of these bounds are best possible. Specializing to the case where \({\mathcal {A}}= {\mathcal {A}}_1 = \cdots = {\mathcal {A}}_k\), we recover a variation of the classical oddtown problem.
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Notes
The sizes of the intersections are zero and nonzero accordingly as opposed to zero and nonzero modulo 2.
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Acknowledgements
The authors were supported by National Science Foundation award DMS-1800332.The authors are very grateful to the referee who carefully read the paper and provided many valuable comments that improved the paper.
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This work was supported by National Science Foundation award DMS-1800332.
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O’Neill, J., Verstraëte, J. A Note on k-Wise Oddtown Problems. Graphs and Combinatorics 38, 101 (2022). https://doi.org/10.1007/s00373-022-02504-z
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DOI: https://doi.org/10.1007/s00373-022-02504-z